The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

division(x, y) → div(x, y, 0)
div(x, y, z) → if(lt(x, y), x, y, inc(z))
if(true, x, y, z) → z
if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
lt(x, 0) → false
lt(0, s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0) → s(0)
inc(s(x)) → s(inc(x))

Rewrite Strategy: INNERMOST

(1) TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to weighted TRS

(2) Obligation:

The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

division(x, y) → div(x, y, 0) [1]
div(x, y, z) → if(lt(x, y), x, y, inc(z)) [1]
if(true, x, y, z) → z [1]
if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z) [1]
minus(x, 0) → x [1]
minus(s(x), s(y)) → minus(x, y) [1]
lt(x, 0) → false [1]
lt(0, s(y)) → true [1]
lt(s(x), s(y)) → lt(x, y) [1]
inc(0) → s(0) [1]
inc(s(x)) → s(inc(x)) [1]

Rewrite Strategy: INNERMOST

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) Obligation:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

division(x, y) → div(x, y, 0) [1]
div(x, y, z) → if(lt(x, y), x, y, inc(z)) [1]
if(true, x, y, z) → z [1]
if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z) [1]
minus(x, 0) → x [1]
minus(s(x), s(y)) → minus(x, y) [1]
lt(x, 0) → false [1]
lt(0, s(y)) → true [1]
lt(s(x), s(y)) → lt(x, y) [1]
inc(0) → s(0) [1]
inc(s(x)) → s(inc(x)) [1]

The TRS has the following type information:

division :: 0:s → 0:s → 0:s div :: 0:s → 0:s → 0:s → 0:s 0 :: 0:s if :: true:false → 0:s → 0:s → 0:s → 0:s lt :: 0:s → 0:s → true:false inc :: 0:s → 0:s true :: true:false false :: true:false s :: 0:s → 0:s minus :: 0:s → 0:s → 0:s

Rewrite Strategy: INNERMOST

(5) CompletionProof (UPPER BOUND(ID) transformation)

The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

division
div
if

(c) The following functions are completely defined:

minus
lt
inc

Due to the following rules being added:

minus(v0, v1) → 0 [0]

And the following fresh constants: none

(6) Obligation:

Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: division(x, y) → div(x, y, 0) [1] div(x, y, z) → if(lt(x, y), x, y, inc(z)) [1] if(true, x, y, z) → z [1] if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z) [1] minus(x, 0) → x [1] minus(s(x), s(y)) → minus(x, y) [1] lt(x, 0) → false [1] lt(0, s(y)) → true [1] lt(s(x), s(y)) → lt(x, y) [1] inc(0) → s(0) [1] inc(s(x)) → s(inc(x)) [1] minus(v0, v1) → 0 [0] The TRS has the following type information: division :: 0:s → 0:s → 0:s div :: 0:s → 0:s → 0:s → 0:s 0 :: 0:s if :: true:false → 0:s → 0:s → 0:s → 0:s lt :: 0:s → 0:s → true:false inc :: 0:s → 0:s true :: true:false false :: true:false s :: 0:s → 0:s minus :: 0:s → 0:s → 0:s Rewrite Strategy: INNERMOST

(7) NarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Narrowed the inner basic terms of all right-hand sides by a single narrowing step.

(8) Obligation:

Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: division(x, y) → div(x, y, 0) [1] div(x, 0, 0) → if(false, x, 0, s(0)) [3] div(x, 0, s(x'')) → if(false, x, 0, s(inc(x''))) [3] div(0, s(y'), 0) → if(true, 0, s(y'), s(0)) [3] div(0, s(y'), s(x1)) → if(true, 0, s(y'), s(inc(x1))) [3] div(s(x'), s(y''), 0) → if(lt(x', y''), s(x'), s(y''), s(0)) [3] div(s(x'), s(y''), s(x2)) → if(lt(x', y''), s(x'), s(y''), s(inc(x2))) [3] if(true, x, y, z) → z [1] if(false, s(x3), s(y), z) → div(minus(x3, y), s(y), z) [2] if(false, x, s(y), z) → div(0, s(y), z) [1] minus(x, 0) → x [1] minus(s(x), s(y)) → minus(x, y) [1] lt(x, 0) → false [1] lt(0, s(y)) → true [1] lt(s(x), s(y)) → lt(x, y) [1] inc(0) → s(0) [1] inc(s(x)) → s(inc(x)) [1] minus(v0, v1) → 0 [0] The TRS has the following type information: division :: 0:s → 0:s → 0:s div :: 0:s → 0:s → 0:s → 0:s 0 :: 0:s if :: true:false → 0:s → 0:s → 0:s → 0:s lt :: 0:s → 0:s → true:false inc :: 0:s → 0:s true :: true:false false :: true:false s :: 0:s → 0:s minus :: 0:s → 0:s → 0:s Rewrite Strategy: INNERMOST

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

0 => 0
true => 1
false => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'', z1) -{ 3 }→ if(lt(x', y''), 1 + x', 1 + y'', 1 + inc(x2)) :|: z' = 1 + x', x' >= 0, y'' >= 0, x2 >= 0, z'' = 1 + y'', z1 = 1 + x2
div(z', z'', z1) -{ 3 }→ if(lt(x', y''), 1 + x', 1 + y'', 1 + 0) :|: z1 = 0, z' = 1 + x', x' >= 0, y'' >= 0, z'' = 1 + y''
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + y', 1 + inc(x1)) :|: x1 >= 0, y' >= 0, z1 = 1 + x1, z' = 0, z'' = 1 + y'
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + y', 1 + 0) :|: z1 = 0, y' >= 0, z' = 0, z'' = 1 + y'
div(z', z'', z1) -{ 3 }→ if(0, x, 0, 1 + inc(x'')) :|: z'' = 0, z' = x, z1 = 1 + x'', x >= 0, x'' >= 0
div(z', z'', z1) -{ 3 }→ if(0, x, 0, 1 + 0) :|: z'' = 0, z1 = 0, z' = x, x >= 0
division(z', z'') -{ 1 }→ div(x, y, 0) :|: z' = x, z'' = y, x >= 0, y >= 0
if(z', z'', z1, z2) -{ 1 }→ z :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
if(z', z'', z1, z2) -{ 2 }→ div(minus(x3, y), 1 + y, z) :|: z >= 0, z'' = 1 + x3, z2 = z, y >= 0, z1 = 1 + y, x3 >= 0, z' = 0
if(z', z'', z1, z2) -{ 1 }→ div(0, 1 + y, z) :|: z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z1 = 1 + y, z' = 0
inc(z') -{ 1 }→ 1 + inc(x) :|: z' = 1 + x, x >= 0
inc(z') -{ 1 }→ 1 + 0 :|: z' = 0
lt(z', z'') -{ 1 }→ lt(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
lt(z', z'') -{ 1 }→ 1 :|: y >= 0, z'' = 1 + y, z' = 0
lt(z', z'') -{ 1 }→ 0 :|: z'' = 0, z' = x, x >= 0
minus(z', z'') -{ 1 }→ x :|: z'' = 0, z' = x, x >= 0
minus(z', z'') -{ 1 }→ minus(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
minus(z', z'') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0

(11) SimplificationProof (BOTH BOUNDS(ID, ID) transformation)

Simplified the RNTS by moving equalities from the constraints into the right-hand sides.

(12) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'', z1) -{ 3 }→ if(lt(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(lt(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + inc(z1 - 1)) :|: z'' = 0, z' >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
division(z', z'') -{ 1 }→ div(z', z'', 0) :|: z' >= 0, z'' >= 0
if(z', z'', z1, z2) -{ 1 }→ z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
if(z', z'', z1, z2) -{ 2 }→ div(minus(z'' - 1, z1 - 1), 1 + (z1 - 1), z2) :|: z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
if(z', z'', z1, z2) -{ 1 }→ div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
inc(z') -{ 1 }→ 1 + inc(z' - 1) :|: z' - 1 >= 0
inc(z') -{ 1 }→ 1 + 0 :|: z' = 0
lt(z', z'') -{ 1 }→ lt(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
lt(z', z'') -{ 1 }→ 1 :|: z'' - 1 >= 0, z' = 0
lt(z', z'') -{ 1 }→ 0 :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 }→ minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
minus(z', z'') -{ 0 }→ 0 :|: z' >= 0, z'' >= 0

(13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID) transformation)

Found the following analysis order by SCC decomposition:

{ lt } { minus } { inc } { div, if } { division }

(14) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'', z1) -{ 3 }→ if(lt(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(lt(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + inc(z1 - 1)) :|: z'' = 0, z' >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
division(z', z'') -{ 1 }→ div(z', z'', 0) :|: z' >= 0, z'' >= 0
if(z', z'', z1, z2) -{ 1 }→ z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
if(z', z'', z1, z2) -{ 2 }→ div(minus(z'' - 1, z1 - 1), 1 + (z1 - 1), z2) :|: z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
if(z', z'', z1, z2) -{ 1 }→ div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
inc(z') -{ 1 }→ 1 + inc(z' - 1) :|: z' - 1 >= 0
inc(z') -{ 1 }→ 1 + 0 :|: z' = 0
lt(z', z'') -{ 1 }→ lt(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
lt(z', z'') -{ 1 }→ 1 :|: z'' - 1 >= 0, z' = 0
lt(z', z'') -{ 1 }→ 0 :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 }→ minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
minus(z', z'') -{ 0 }→ 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {lt}, {minus}, {inc}, {div,if}, {division}

(15) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: lt
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(16) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'', z1) -{ 3 }→ if(lt(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(lt(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + inc(z1 - 1)) :|: z'' = 0, z' >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
division(z', z'') -{ 1 }→ div(z', z'', 0) :|: z' >= 0, z'' >= 0
if(z', z'', z1, z2) -{ 1 }→ z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
if(z', z'', z1, z2) -{ 2 }→ div(minus(z'' - 1, z1 - 1), 1 + (z1 - 1), z2) :|: z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
if(z', z'', z1, z2) -{ 1 }→ div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
inc(z') -{ 1 }→ 1 + inc(z' - 1) :|: z' - 1 >= 0
inc(z') -{ 1 }→ 1 + 0 :|: z' = 0
lt(z', z'') -{ 1 }→ lt(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
lt(z', z'') -{ 1 }→ 1 :|: z'' - 1 >= 0, z' = 0
lt(z', z'') -{ 1 }→ 0 :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 }→ minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
minus(z', z'') -{ 0 }→ 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {lt}, {minus}, {inc}, {div,if}, {division}
Previous analysis results are:

lt: runtime: ?, size: O(1) [1]

(17) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using PUBS for: lt
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z''

(18) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'', z1) -{ 3 }→ if(lt(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(lt(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + inc(z1 - 1)) :|: z'' = 0, z' >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
division(z', z'') -{ 1 }→ div(z', z'', 0) :|: z' >= 0, z'' >= 0
if(z', z'', z1, z2) -{ 1 }→ z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
if(z', z'', z1, z2) -{ 2 }→ div(minus(z'' - 1, z1 - 1), 1 + (z1 - 1), z2) :|: z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
if(z', z'', z1, z2) -{ 1 }→ div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
inc(z') -{ 1 }→ 1 + inc(z' - 1) :|: z' - 1 >= 0
inc(z') -{ 1 }→ 1 + 0 :|: z' = 0
lt(z', z'') -{ 1 }→ lt(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
lt(z', z'') -{ 1 }→ 1 :|: z'' - 1 >= 0, z' = 0
lt(z', z'') -{ 1 }→ 0 :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 }→ minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
minus(z', z'') -{ 0 }→ 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {minus}, {inc}, {div,if}, {division}
Previous analysis results are:

lt: runtime: O(n1) [1 + z''], size: O(1) [1]

(19) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(20) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'', z1) -{ 3 + z'' }→ if(s, 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
div(z', z'', z1) -{ 3 + z'' }→ if(s', 1 + (z' - 1), 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + inc(z1 - 1)) :|: z'' = 0, z' >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
division(z', z'') -{ 1 }→ div(z', z'', 0) :|: z' >= 0, z'' >= 0
if(z', z'', z1, z2) -{ 1 }→ z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
if(z', z'', z1, z2) -{ 2 }→ div(minus(z'' - 1, z1 - 1), 1 + (z1 - 1), z2) :|: z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
if(z', z'', z1, z2) -{ 1 }→ div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
inc(z') -{ 1 }→ 1 + inc(z' - 1) :|: z' - 1 >= 0
inc(z') -{ 1 }→ 1 + 0 :|: z' = 0
lt(z', z'') -{ 1 + z'' }→ s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
lt(z', z'') -{ 1 }→ 1 :|: z'' - 1 >= 0, z' = 0
lt(z', z'') -{ 1 }→ 0 :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 }→ minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
minus(z', z'') -{ 0 }→ 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {minus}, {inc}, {div,if}, {division}
Previous analysis results are:

lt: runtime: O(n1) [1 + z''], size: O(1) [1]

(21) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using KoAT for: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z'

(22) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'', z1) -{ 3 + z'' }→ if(s, 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
div(z', z'', z1) -{ 3 + z'' }→ if(s', 1 + (z' - 1), 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + inc(z1 - 1)) :|: z'' = 0, z' >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
division(z', z'') -{ 1 }→ div(z', z'', 0) :|: z' >= 0, z'' >= 0
if(z', z'', z1, z2) -{ 1 }→ z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
if(z', z'', z1, z2) -{ 2 }→ div(minus(z'' - 1, z1 - 1), 1 + (z1 - 1), z2) :|: z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
if(z', z'', z1, z2) -{ 1 }→ div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
inc(z') -{ 1 }→ 1 + inc(z' - 1) :|: z' - 1 >= 0
inc(z') -{ 1 }→ 1 + 0 :|: z' = 0
lt(z', z'') -{ 1 + z'' }→ s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
lt(z', z'') -{ 1 }→ 1 :|: z'' - 1 >= 0, z' = 0
lt(z', z'') -{ 1 }→ 0 :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 }→ minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
minus(z', z'') -{ 0 }→ 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {minus}, {inc}, {div,if}, {division}
Previous analysis results are:

lt: runtime: O(n1) [1 + z''], size: O(1) [1] minus: runtime: ?, size: O(n1) [z']

(23) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z''

(24) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'', z1) -{ 3 + z'' }→ if(s, 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
div(z', z'', z1) -{ 3 + z'' }→ if(s', 1 + (z' - 1), 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + inc(z1 - 1)) :|: z'' = 0, z' >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
division(z', z'') -{ 1 }→ div(z', z'', 0) :|: z' >= 0, z'' >= 0
if(z', z'', z1, z2) -{ 1 }→ z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
if(z', z'', z1, z2) -{ 2 }→ div(minus(z'' - 1, z1 - 1), 1 + (z1 - 1), z2) :|: z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
if(z', z'', z1, z2) -{ 1 }→ div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
inc(z') -{ 1 }→ 1 + inc(z' - 1) :|: z' - 1 >= 0
inc(z') -{ 1 }→ 1 + 0 :|: z' = 0
lt(z', z'') -{ 1 + z'' }→ s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
lt(z', z'') -{ 1 }→ 1 :|: z'' - 1 >= 0, z' = 0
lt(z', z'') -{ 1 }→ 0 :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 }→ minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
minus(z', z'') -{ 0 }→ 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {inc}, {div,if}, {division}
Previous analysis results are:

lt: runtime: O(n1) [1 + z''], size: O(1) [1] minus: runtime: O(n1) [1 + z''], size: O(n1) [z']

(25) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(26) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'', z1) -{ 3 + z'' }→ if(s, 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
div(z', z'', z1) -{ 3 + z'' }→ if(s', 1 + (z' - 1), 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + inc(z1 - 1)) :|: z'' = 0, z' >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
division(z', z'') -{ 1 }→ div(z', z'', 0) :|: z' >= 0, z'' >= 0
if(z', z'', z1, z2) -{ 1 }→ z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
if(z', z'', z1, z2) -{ 2 + z1 }→ div(s1, 1 + (z1 - 1), z2) :|: s1 >= 0, s1 <= 1 * (z'' - 1), z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
if(z', z'', z1, z2) -{ 1 }→ div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
inc(z') -{ 1 }→ 1 + inc(z' - 1) :|: z' - 1 >= 0
inc(z') -{ 1 }→ 1 + 0 :|: z' = 0
lt(z', z'') -{ 1 + z'' }→ s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
lt(z', z'') -{ 1 }→ 1 :|: z'' - 1 >= 0, z' = 0
lt(z', z'') -{ 1 }→ 0 :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 + z'' }→ s2 :|: s2 >= 0, s2 <= 1 * (z' - 1), z' - 1 >= 0, z'' - 1 >= 0
minus(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
minus(z', z'') -{ 0 }→ 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {inc}, {div,if}, {division}
Previous analysis results are:

lt: runtime: O(n1) [1 + z''], size: O(1) [1] minus: runtime: O(n1) [1 + z''], size: O(n1) [z']

(27) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: inc
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z'

(28) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'', z1) -{ 3 + z'' }→ if(s, 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
div(z', z'', z1) -{ 3 + z'' }→ if(s', 1 + (z' - 1), 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + inc(z1 - 1)) :|: z'' = 0, z' >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
division(z', z'') -{ 1 }→ div(z', z'', 0) :|: z' >= 0, z'' >= 0
if(z', z'', z1, z2) -{ 1 }→ z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
if(z', z'', z1, z2) -{ 2 + z1 }→ div(s1, 1 + (z1 - 1), z2) :|: s1 >= 0, s1 <= 1 * (z'' - 1), z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
if(z', z'', z1, z2) -{ 1 }→ div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
inc(z') -{ 1 }→ 1 + inc(z' - 1) :|: z' - 1 >= 0
inc(z') -{ 1 }→ 1 + 0 :|: z' = 0
lt(z', z'') -{ 1 + z'' }→ s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
lt(z', z'') -{ 1 }→ 1 :|: z'' - 1 >= 0, z' = 0
lt(z', z'') -{ 1 }→ 0 :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 + z'' }→ s2 :|: s2 >= 0, s2 <= 1 * (z' - 1), z' - 1 >= 0, z'' - 1 >= 0
minus(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
minus(z', z'') -{ 0 }→ 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {inc}, {div,if}, {division}
Previous analysis results are:

lt: runtime: O(n1) [1 + z''], size: O(1) [1] minus: runtime: O(n1) [1 + z''], size: O(n1) [z'] inc: runtime: ?, size: O(n1) [1 + z']

(29) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: inc
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z'

(30) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'', z1) -{ 3 + z'' }→ if(s, 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
div(z', z'', z1) -{ 3 + z'' }→ if(s', 1 + (z' - 1), 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + inc(z1 - 1)) :|: z'' = 0, z' >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
division(z', z'') -{ 1 }→ div(z', z'', 0) :|: z' >= 0, z'' >= 0
if(z', z'', z1, z2) -{ 1 }→ z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
if(z', z'', z1, z2) -{ 2 + z1 }→ div(s1, 1 + (z1 - 1), z2) :|: s1 >= 0, s1 <= 1 * (z'' - 1), z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
if(z', z'', z1, z2) -{ 1 }→ div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
inc(z') -{ 1 }→ 1 + inc(z' - 1) :|: z' - 1 >= 0
inc(z') -{ 1 }→ 1 + 0 :|: z' = 0
lt(z', z'') -{ 1 + z'' }→ s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
lt(z', z'') -{ 1 }→ 1 :|: z'' - 1 >= 0, z' = 0
lt(z', z'') -{ 1 }→ 0 :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 + z'' }→ s2 :|: s2 >= 0, s2 <= 1 * (z' - 1), z' - 1 >= 0, z'' - 1 >= 0
minus(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
minus(z', z'') -{ 0 }→ 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {div,if}, {division}
Previous analysis results are:

lt: runtime: O(n1) [1 + z''], size: O(1) [1] minus: runtime: O(n1) [1 + z''], size: O(n1) [z'] inc: runtime: O(n1) [1 + z'], size: O(n1) [1 + z']

(31) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(32) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'', z1) -{ 3 + z'' }→ if(s, 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
div(z', z'', z1) -{ 3 + z'' + z1 }→ if(s', 1 + (z' - 1), 1 + (z'' - 1), 1 + s5) :|: s5 >= 0, s5 <= 1 * (z1 - 1) + 1, s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 + z1 }→ if(1, 0, 1 + (z'' - 1), 1 + s4) :|: s4 >= 0, s4 <= 1 * (z1 - 1) + 1, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 + z1 }→ if(0, z', 0, 1 + s3) :|: s3 >= 0, s3 <= 1 * (z1 - 1) + 1, z'' = 0, z' >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
division(z', z'') -{ 1 }→ div(z', z'', 0) :|: z' >= 0, z'' >= 0
if(z', z'', z1, z2) -{ 1 }→ z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
if(z', z'', z1, z2) -{ 2 + z1 }→ div(s1, 1 + (z1 - 1), z2) :|: s1 >= 0, s1 <= 1 * (z'' - 1), z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
if(z', z'', z1, z2) -{ 1 }→ div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
inc(z') -{ 1 + z' }→ 1 + s6 :|: s6 >= 0, s6 <= 1 * (z' - 1) + 1, z' - 1 >= 0
inc(z') -{ 1 }→ 1 + 0 :|: z' = 0
lt(z', z'') -{ 1 + z'' }→ s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
lt(z', z'') -{ 1 }→ 1 :|: z'' - 1 >= 0, z' = 0
lt(z', z'') -{ 1 }→ 0 :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 + z'' }→ s2 :|: s2 >= 0, s2 <= 1 * (z' - 1), z' - 1 >= 0, z'' - 1 >= 0
minus(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
minus(z', z'') -{ 0 }→ 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {div,if}, {division}
Previous analysis results are:

lt: runtime: O(n1) [1 + z''], size: O(1) [1] minus: runtime: O(n1) [1 + z''], size: O(n1) [z'] inc: runtime: O(n1) [1 + z'], size: O(n1) [1 + z']

(33) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using PUBS for: div
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z' + z1

Computed SIZE bound using CoFloCo for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z'' + z2

(34) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'', z1) -{ 3 + z'' }→ if(s, 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
div(z', z'', z1) -{ 3 + z'' + z1 }→ if(s', 1 + (z' - 1), 1 + (z'' - 1), 1 + s5) :|: s5 >= 0, s5 <= 1 * (z1 - 1) + 1, s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 + z1 }→ if(1, 0, 1 + (z'' - 1), 1 + s4) :|: s4 >= 0, s4 <= 1 * (z1 - 1) + 1, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 + z1 }→ if(0, z', 0, 1 + s3) :|: s3 >= 0, s3 <= 1 * (z1 - 1) + 1, z'' = 0, z' >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
division(z', z'') -{ 1 }→ div(z', z'', 0) :|: z' >= 0, z'' >= 0
if(z', z'', z1, z2) -{ 1 }→ z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
if(z', z'', z1, z2) -{ 2 + z1 }→ div(s1, 1 + (z1 - 1), z2) :|: s1 >= 0, s1 <= 1 * (z'' - 1), z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
if(z', z'', z1, z2) -{ 1 }→ div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
inc(z') -{ 1 + z' }→ 1 + s6 :|: s6 >= 0, s6 <= 1 * (z' - 1) + 1, z' - 1 >= 0
inc(z') -{ 1 }→ 1 + 0 :|: z' = 0
lt(z', z'') -{ 1 + z'' }→ s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
lt(z', z'') -{ 1 }→ 1 :|: z'' - 1 >= 0, z' = 0
lt(z', z'') -{ 1 }→ 0 :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 + z'' }→ s2 :|: s2 >= 0, s2 <= 1 * (z' - 1), z' - 1 >= 0, z'' - 1 >= 0
minus(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
minus(z', z'') -{ 0 }→ 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {div,if}, {division}
Previous analysis results are:

lt: runtime: O(n1) [1 + z''], size: O(1) [1] minus: runtime: O(n1) [1 + z''], size: O(n1) [z'] inc: runtime: O(n1) [1 + z'], size: O(n1) [1 + z'] div: runtime: ?, size: O(n1) [1 + z' + z1] if: runtime: ?, size: O(n1) [1 + z'' + z2]

(35) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using PUBS for: div
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 5 + 7·z' + 2·z'·z'' + z'·z1 + z'² + z'' + z1

Computed RUNTIME bound using PUBS for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 6 + 5·z'' + 2·z''·z1 + z''·z2 + z''² + z1 + z2

(36) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'', z1) -{ 3 + z'' }→ if(s, 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
div(z', z'', z1) -{ 3 + z'' + z1 }→ if(s', 1 + (z' - 1), 1 + (z'' - 1), 1 + s5) :|: s5 >= 0, s5 <= 1 * (z1 - 1) + 1, s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 + z1 }→ if(1, 0, 1 + (z'' - 1), 1 + s4) :|: s4 >= 0, s4 <= 1 * (z1 - 1) + 1, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 }→ if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 3 + z1 }→ if(0, z', 0, 1 + s3) :|: s3 >= 0, s3 <= 1 * (z1 - 1) + 1, z'' = 0, z' >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 3 }→ if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
division(z', z'') -{ 1 }→ div(z', z'', 0) :|: z' >= 0, z'' >= 0
if(z', z'', z1, z2) -{ 1 }→ z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
if(z', z'', z1, z2) -{ 2 + z1 }→ div(s1, 1 + (z1 - 1), z2) :|: s1 >= 0, s1 <= 1 * (z'' - 1), z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
if(z', z'', z1, z2) -{ 1 }→ div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
inc(z') -{ 1 + z' }→ 1 + s6 :|: s6 >= 0, s6 <= 1 * (z' - 1) + 1, z' - 1 >= 0
inc(z') -{ 1 }→ 1 + 0 :|: z' = 0
lt(z', z'') -{ 1 + z'' }→ s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
lt(z', z'') -{ 1 }→ 1 :|: z'' - 1 >= 0, z' = 0
lt(z', z'') -{ 1 }→ 0 :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 + z'' }→ s2 :|: s2 >= 0, s2 <= 1 * (z' - 1), z' - 1 >= 0, z'' - 1 >= 0
minus(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
minus(z', z'') -{ 0 }→ 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {division}
Previous analysis results are:

lt: runtime: O(n1) [1 + z''], size: O(1) [1] minus: runtime: O(n1) [1 + z''], size: O(n1) [z'] inc: runtime: O(n1) [1 + z'], size: O(n1) [1 + z'] div: runtime: O(n2) [5 + 7·z' + 2·z'·z'' + z'·z1 + z'2 + z'' + z1], size: O(n1) [1 + z' + z1] if: runtime: O(n2) [6 + 5·z'' + 2·z''·z1 + z''·z2 + z''2 + z1 + z2], size: O(n1) [1 + z'' + z2]

(37) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(38) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'', z1) -{ 10 + z'' }→ s10 :|: s10 >= 0, s10 <= 1 * (1 + 0) + 1 + 1 * 0, z1 = 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 10 + s4 + z'' + z1 }→ s11 :|: s11 >= 0, s11 <= 1 * (1 + s4) + 1 + 1 * 0, s4 >= 0, s4 <= 1 * (z1 - 1) + 1, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 10 + 6·z' + 2·z'·z'' + z'² + 2·z'' }→ s12 :|: s12 >= 0, s12 <= 1 * (1 + 0) + 1 + 1 * (1 + (z' - 1)), s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
div(z', z'', z1) -{ 10 + s5 + s5·z' + 6·z' + 2·z'·z'' + z'² + 2·z'' + z1 }→ s13 :|: s13 >= 0, s13 <= 1 * (1 + s5) + 1 + 1 * (1 + (z' - 1)), s5 >= 0, s5 <= 1 * (z1 - 1) + 1, s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 10 + 6·z' + z'² }→ s8 :|: s8 >= 0, s8 <= 1 * (1 + 0) + 1 + 1 * z', z'' = 0, z1 = 0, z' >= 0
div(z', z'', z1) -{ 10 + s3 + s3·z' + 6·z' + z'² + z1 }→ s9 :|: s9 >= 0, s9 <= 1 * (1 + s3) + 1 + 1 * z', s3 >= 0, s3 <= 1 * (z1 - 1) + 1, z'' = 0, z' >= 0, z1 - 1 >= 0
division(z', z'') -{ 6 + 7·z' + 2·z'·z'' + z'² + z'' }→ s7 :|: s7 >= 0, s7 <= 1 + 1 * z' + 1 * 0, z' >= 0, z'' >= 0
if(z', z'', z1, z2) -{ 7 + 7·s1 + 2·s1·z1 + s1·z2 + s1² + 2·z1 + z2 }→ s14 :|: s14 >= 0, s14 <= 1 + 1 * s1 + 1 * z2, s1 >= 0, s1 <= 1 * (z'' - 1), z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
if(z', z'', z1, z2) -{ 6 + z1 + z2 }→ s15 :|: s15 >= 0, s15 <= 1 + 1 * 0 + 1 * z2, z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
if(z', z'', z1, z2) -{ 1 }→ z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
inc(z') -{ 1 + z' }→ 1 + s6 :|: s6 >= 0, s6 <= 1 * (z' - 1) + 1, z' - 1 >= 0
inc(z') -{ 1 }→ 1 + 0 :|: z' = 0
lt(z', z'') -{ 1 + z'' }→ s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
lt(z', z'') -{ 1 }→ 1 :|: z'' - 1 >= 0, z' = 0
lt(z', z'') -{ 1 }→ 0 :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 + z'' }→ s2 :|: s2 >= 0, s2 <= 1 * (z' - 1), z' - 1 >= 0, z'' - 1 >= 0
minus(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
minus(z', z'') -{ 0 }→ 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {division}
Previous analysis results are:

lt: runtime: O(n1) [1 + z''], size: O(1) [1] minus: runtime: O(n1) [1 + z''], size: O(n1) [z'] inc: runtime: O(n1) [1 + z'], size: O(n1) [1 + z'] div: runtime: O(n2) [5 + 7·z' + 2·z'·z'' + z'·z1 + z'2 + z'' + z1], size: O(n1) [1 + z' + z1] if: runtime: O(n2) [6 + 5·z'' + 2·z''·z1 + z''·z2 + z''2 + z1 + z2], size: O(n1) [1 + z'' + z2]

(39) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: division
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z'

(40) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'', z1) -{ 10 + z'' }→ s10 :|: s10 >= 0, s10 <= 1 * (1 + 0) + 1 + 1 * 0, z1 = 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 10 + s4 + z'' + z1 }→ s11 :|: s11 >= 0, s11 <= 1 * (1 + s4) + 1 + 1 * 0, s4 >= 0, s4 <= 1 * (z1 - 1) + 1, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 10 + 6·z' + 2·z'·z'' + z'² + 2·z'' }→ s12 :|: s12 >= 0, s12 <= 1 * (1 + 0) + 1 + 1 * (1 + (z' - 1)), s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
div(z', z'', z1) -{ 10 + s5 + s5·z' + 6·z' + 2·z'·z'' + z'² + 2·z'' + z1 }→ s13 :|: s13 >= 0, s13 <= 1 * (1 + s5) + 1 + 1 * (1 + (z' - 1)), s5 >= 0, s5 <= 1 * (z1 - 1) + 1, s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 10 + 6·z' + z'² }→ s8 :|: s8 >= 0, s8 <= 1 * (1 + 0) + 1 + 1 * z', z'' = 0, z1 = 0, z' >= 0
div(z', z'', z1) -{ 10 + s3 + s3·z' + 6·z' + z'² + z1 }→ s9 :|: s9 >= 0, s9 <= 1 * (1 + s3) + 1 + 1 * z', s3 >= 0, s3 <= 1 * (z1 - 1) + 1, z'' = 0, z' >= 0, z1 - 1 >= 0
division(z', z'') -{ 6 + 7·z' + 2·z'·z'' + z'² + z'' }→ s7 :|: s7 >= 0, s7 <= 1 + 1 * z' + 1 * 0, z' >= 0, z'' >= 0
if(z', z'', z1, z2) -{ 7 + 7·s1 + 2·s1·z1 + s1·z2 + s1² + 2·z1 + z2 }→ s14 :|: s14 >= 0, s14 <= 1 + 1 * s1 + 1 * z2, s1 >= 0, s1 <= 1 * (z'' - 1), z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
if(z', z'', z1, z2) -{ 6 + z1 + z2 }→ s15 :|: s15 >= 0, s15 <= 1 + 1 * 0 + 1 * z2, z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
if(z', z'', z1, z2) -{ 1 }→ z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
inc(z') -{ 1 + z' }→ 1 + s6 :|: s6 >= 0, s6 <= 1 * (z' - 1) + 1, z' - 1 >= 0
inc(z') -{ 1 }→ 1 + 0 :|: z' = 0
lt(z', z'') -{ 1 + z'' }→ s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
lt(z', z'') -{ 1 }→ 1 :|: z'' - 1 >= 0, z' = 0
lt(z', z'') -{ 1 }→ 0 :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 + z'' }→ s2 :|: s2 >= 0, s2 <= 1 * (z' - 1), z' - 1 >= 0, z'' - 1 >= 0
minus(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
minus(z', z'') -{ 0 }→ 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {division}
Previous analysis results are:

lt: runtime: O(n1) [1 + z''], size: O(1) [1] minus: runtime: O(n1) [1 + z''], size: O(n1) [z'] inc: runtime: O(n1) [1 + z'], size: O(n1) [1 + z'] div: runtime: O(n2) [5 + 7·z' + 2·z'·z'' + z'·z1 + z'2 + z'' + z1], size: O(n1) [1 + z' + z1] if: runtime: O(n2) [6 + 5·z'' + 2·z''·z1 + z''·z2 + z''2 + z1 + z2], size: O(n1) [1 + z'' + z2] division: runtime: ?, size: O(n1) [1 + z']

(41) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using KoAT for: division
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 6 + 7·z' + 2·z'·z'' + z'² + z''

(42) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'', z1) -{ 10 + z'' }→ s10 :|: s10 >= 0, s10 <= 1 * (1 + 0) + 1 + 1 * 0, z1 = 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 10 + s4 + z'' + z1 }→ s11 :|: s11 >= 0, s11 <= 1 * (1 + s4) + 1 + 1 * 0, s4 >= 0, s4 <= 1 * (z1 - 1) + 1, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
div(z', z'', z1) -{ 10 + 6·z' + 2·z'·z'' + z'² + 2·z'' }→ s12 :|: s12 >= 0, s12 <= 1 * (1 + 0) + 1 + 1 * (1 + (z' - 1)), s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
div(z', z'', z1) -{ 10 + s5 + s5·z' + 6·z' + 2·z'·z'' + z'² + 2·z'' + z1 }→ s13 :|: s13 >= 0, s13 <= 1 * (1 + s5) + 1 + 1 * (1 + (z' - 1)), s5 >= 0, s5 <= 1 * (z1 - 1) + 1, s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
div(z', z'', z1) -{ 10 + 6·z' + z'² }→ s8 :|: s8 >= 0, s8 <= 1 * (1 + 0) + 1 + 1 * z', z'' = 0, z1 = 0, z' >= 0
div(z', z'', z1) -{ 10 + s3 + s3·z' + 6·z' + z'² + z1 }→ s9 :|: s9 >= 0, s9 <= 1 * (1 + s3) + 1 + 1 * z', s3 >= 0, s3 <= 1 * (z1 - 1) + 1, z'' = 0, z' >= 0, z1 - 1 >= 0
division(z', z'') -{ 6 + 7·z' + 2·z'·z'' + z'² + z'' }→ s7 :|: s7 >= 0, s7 <= 1 + 1 * z' + 1 * 0, z' >= 0, z'' >= 0
if(z', z'', z1, z2) -{ 7 + 7·s1 + 2·s1·z1 + s1·z2 + s1² + 2·z1 + z2 }→ s14 :|: s14 >= 0, s14 <= 1 + 1 * s1 + 1 * z2, s1 >= 0, s1 <= 1 * (z'' - 1), z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
if(z', z'', z1, z2) -{ 6 + z1 + z2 }→ s15 :|: s15 >= 0, s15 <= 1 + 1 * 0 + 1 * z2, z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
if(z', z'', z1, z2) -{ 1 }→ z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
inc(z') -{ 1 + z' }→ 1 + s6 :|: s6 >= 0, s6 <= 1 * (z' - 1) + 1, z' - 1 >= 0
inc(z') -{ 1 }→ 1 + 0 :|: z' = 0
lt(z', z'') -{ 1 + z'' }→ s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
lt(z', z'') -{ 1 }→ 1 :|: z'' - 1 >= 0, z' = 0
lt(z', z'') -{ 1 }→ 0 :|: z'' = 0, z' >= 0
minus(z', z'') -{ 1 + z'' }→ s2 :|: s2 >= 0, s2 <= 1 * (z' - 1), z' - 1 >= 0, z'' - 1 >= 0
minus(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
minus(z', z'') -{ 0 }→ 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed:
Previous analysis results are:

lt: runtime: O(n1) [1 + z''], size: O(1) [1] minus: runtime: O(n1) [1 + z''], size: O(n1) [z'] inc: runtime: O(n1) [1 + z'], size: O(n1) [1 + z'] div: runtime: O(n2) [5 + 7·z' + 2·z'·z'' + z'·z1 + z'2 + z'' + z1], size: O(n1) [1 + z' + z1] if: runtime: O(n2) [6 + 5·z'' + 2·z''·z1 + z''·z2 + z''2 + z1 + z2], size: O(n1) [1 + z'' + z2] division: runtime: O(n2) [6 + 7·z' + 2·z'·z'' + z'2 + z''], size: O(n1) [1 + z']

(43) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(44) BOUNDS(1, n^2)